Fluid analog models for gravity are based on the idea that any spacetime geometry admits a reinterpretation in which space is thought of as a fluid flowing with a prescribed velocity. This fluid picture is a restatement of the ADM decomposition of the metric. Most of the literature has focused on flat spatial geometries and physical fluid flows, with a view toward possible laboratory realizations. Here we relax these conditions and consider fluid flows on curved and time-dependent spatial geometries, as a way of understanding and visualizing solutions to general relativity. We illustrate the utility of the approach with rotating black holes. For the Kerr black hole we develop a fluid description based on Doran coordinates. For spinning BTZ black holes we develop two different fluid descriptions. One involves static conical spatial slices, with the fluid orbiting the tip of the cone. The other resembles a cosmology, with the fluid flowing on a time-dependent cylindrical geometry.